The Vieta-Jumping method. Three problems of number theory

      An article that sets out the method can be found here. It was published by YIMIN GE in the journal Mathematical Reflections, 5, (2007). See also here. And here for a book.

 

          PROBLEM #1 (IMO 1988, Problem 6) 

          Let $a,b$ be positive integers so that $ab+1$ divides $a^2+b^2$. Prove that $\frac{a^2+b^2}{ab+1}$ is a perfect square.

          PROBLEM #2

          Let $x,y$ be positive integers so that $xy$ divides $x^2+y^2+1$. Prove that

$$\frac{x^2+y^2+1}{xy}=3.$$

 

           PROBLEM #3 (IMO 2007, Problem 5)

           Let $a,b$ be positive integers. Show that if $4ab-1$ divides $(4a^2-1)^2$, then $a=b$.